Appropriate offerings will vary from deity to deity. Gods have personal tastes just as we do. If you're going to be having a dinner guest, and you know they don't like peas yet you serve them, how rude is that? You're not likely to develop a relationship that way. It can be worse; serving pork to an observant Jew isn't just offending, it's an offense. The same is true with deities. Some are offended by certain offerings (most commonly meat, which is particularly bothersome to some domestic deities).
It is sometimes possible to find out what to offer through research. Cultures often have traditions: water mixed with wine for Greek deities, sake for Japanese kami, etc. Individual deities may have preferences. For instance, we know from Apuleius (Golden Ass 3.27) that Epona liked roses. Other times, you will need to experiment, starting with what seems best and seeing what happens.
There are some patterns linking deity type and offerings that help with this. The following list gives some of them. They are not meant to be taken as hard-and-fast rules—a deity from a culture that does not drink milk might be confused by a milk offering—but they are a point from which to start experimenting.
Ancestors: Food and drink from the family table, bread (especially dark), beer, legumes, caraway, hair. If you are offering to a particular Ancestor, then use their favorite foods.
Birth deities: Bread, eggs, cookies, sandalwood, mint, roses, hair from a first haircut.
Border guardians: Pigs, eggs, honey, cakes, milk, wine, flower garlands.
Culture deities: Songs, poems, prepared food, articles of culture such as handiworks.
Death deities: Pork, dogs, beer. This is one exception to the rule of sharing; you don't want to share with death. Make the offering, but don't consume any part of it.
Deities in general: Bread, butter or oil (burned), wine, beer, mead; incense (particularly frankincense).
Earth deities: Bread, pigs, beer.
Fire deities: Flammable items, especially liquids such as vegetable oils, melted clarified butter, whiskey, or other distilled liquors. (Note: Be careful with distilled drinks. Make sure you pour safely, either with a long spoon or while wearing a fire glove, since they will flare up. Also, when the alcohol and any sugar are all burned, you're left with essentially water, so don't pour too much at a time, or you will extinguish the fire).
Garden Spirits: Bread, grain, fruit, water, milk, honey, bay leaf incense, flowers.
Hearth guardians: Bread, butter (especially if clarified), milk, pine incense, rosemary. Generally not meat.
House Spirits: Bread, salt, wine, milk, food from the family table, butter, wine, beer, frankincense, rosemary.
Lunar deities: Milk, white bread, white flowers, silver.
Nature Spirits (in general): The local grain (cornmeal in America, wheat or barley in England, etc.), bread, cheese, sage, beer, flowers, shiny things.
Nature Spirits (American): Cornmeal, sage, tobacco, shiny things.
Solar deities: Horses, white wine, mead, gold, butter.
Storm deities: Cattle, beer, meteorites, sledgehammers, axes, flint, stone tools.
Technology deities: Tools, products of labor.
Threshold guardians: Barley, bread, wine, juniper.
War deities: Cattle, goats, iron, weapons.